Google Music Playlist Duplicate Songs/Nonsyncying Playlist Solution
For those of you who have had these problems (which is a number of you, as I found out after a quick search), I had the same problems and found a solution after tinkering about. The alternative, deleting all old music and resyncing, often still gave me problems so I'm hoping this will be helpful. If anyone has found a more elegant solution, please share it.
Step 1: Create a NEW Playlist in iTunes WITH A UNIQUE NAME THAT YOU HAVE NEVER USED BEFORE
This is important. For whatever reason, the music player will assume a playlist with a name you've used in the past is already synced and it will ignore it. It also has to be a NEW playlist, not an old playlist that you rename. I used "Test1", "Test2", etc. Don't worry, you'll be able to change it to something else later.
Step 2: Add one song to the playlist that you do not actually want in the playlist.
Again, important. When you sync in the next step, the playlist that appears in google music won't contain the song you added.
Step 3: It may take a minute or so for the playlist to appear in the list of playlists to sync; wait until it is visible and then sync the playlist with google music manager.
Step 4: Wait for the Playlist to appear in google music. It will appear as an empty playlist.
Step 5: Remove the song from the playlist in iTunes and wait for the playlist to disappear from the list of playlists to sync in google music manager before syncing the playlists.
Empty playlists are not recognized as syncable so they will eventually disappear from the list of playlists that you can sync.
Step 6: Add ANOTHER song you do not actually want in the playlist
Again, when you resync in the next step, the song will not appear in the playlist on google music.
Step 7: Wait for the playlist to appear in google music manager, and then resync.
Step 8: After the playlist appears again in google music, add all the songs you actually want in the playlists, and sync the music manager.
You might have to sync a couple times and/or wait a couple minutes, but eventually, you will see that the songs are being added to the playlist in google music.
Step 9: After your songs are added, you can rename the playlist both in iTunes and in Google Music. It will still sync properly if you add and/or remove songs from it.